Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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Inverse Laplace Transform of $ \left(\frac{1-s^{1/2}}{s^2}\right)^2$

I found this question in my N.P Bali's Engineering Mathematics 7th Edition. I could not find any solved questions related to this. How can I find the Inverse Laplace Transform of : ...
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49 views

How to calculate the inverse of the line integeral.

Let $f$ be a polynomial function, $$ f(x) = a_0 + a_1 x + ... + a_d x^d $$ where $a_0$, $a_1$, ..., $a_d$ are parameters and usually $d \le 6$. Let $g$ be the line integral of $f$, $$ g(x) = ...
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68 views

Double Think about Numerosity

According to standard mathematics, the Natural Numbers are given. Moreover, they are given as a (completed) Infinite Set. This set is commonly denoted as: $$ \mathbb{N} = \left\{ ...
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Inverse function (basic algbra math)

Consider the following function: $f(x) = {1 / (x-6) }$ Find a formula for the inverse of the function. Here is what have so far? $y = 1/(x-6)$ ---> $ x = 1/(y-6) $ But my embarrassing problem is ...
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Inverse function of $f(t)=5 +\frac{75}{1 + e^{-((t-50)/10)}}$

i need to find the inverse function of $$ v= f(t)=5 + \frac{75}{1 + e^{-\frac{t-50}{10}}} $$ so far i have $$ v - 5 = \frac{75}{1 + e^{-\frac{t-50}{10}}} $$ $$ (v-5) \left(1 + ...
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24 views

Show: $f\colon X\to Y$ bijective $\Longleftrightarrow$ f has an inverse function

As the title says, I would like to prove that $f\colon X\to Y$ bijective $\Longleftrightarrow$ f has an inverse function. Proof $\Rightarrow$ Let $f$ be bijective. That means $\forall y\in ...
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34 views

PID question in Ireland and Rosen

Context: In Ireland and Rosen's 'A classic introduction to number theory' on page 11, the proof that in a PID$=R$, there is an integer $n$ such that, for a prime $p$ and any $b\in R$, $p^n \mid b , ...
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3answers
57 views

Algebra question: Finding inverse function

This question is about finding the inverse function of $f(x)=-\sqrt{9-x^2}$ I seem to be making an error with one of the manipulations. Here is my attempt. $$x=-\sqrt{9-y^2}$$ ...
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57 views

Find the inverse of $f(x) = (x+1)/(x-8)$

Find the inverse of this function: I have gotten this far: $x = y+1/y-8$ $x(y-8) = y+1$ $x(y-8)-1=y$ $xy-8x - 1 = y$ I think I went backwards?
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How to find the inverse of $f(x) = \frac{x+2}x$?

What approach would be ideal in finding the inverse of $f(x) = \frac{x+2}x$?
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40 views

The inverse of a transpose matrix to “cancel” the transpose?

When it comes to solving and equation containing matrices I don't always understand some of the rules involved. In particular, I am trying to figure out the derivation of the Gauss-Newton algorithm. ...
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40 views

Inverse functions determination by integral

From "Inverse functions and differentiation": Integrating this relationship gives $$ f^{-1}(x)=\int\frac{1}{f'(f^{-1}(x))}\,dx + c. $$ This is only useful if the integral exists. ...
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18 views

Possible to find inverse or eigenvalues of a block diagonal matrix with upper and lower diagonal matrices

I just encountered a matrix problem of finding inverse of eigenvalues of a block diagonal matrix with upper and lower also matrices of the form where A and B are full rank matrices. Is there any ...
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45 views

prove function is surjective /analysis proofs!!

Suppose $f:(a,b)\longrightarrow\mathbb R$, differentiable, where $(a,b)\subseteq\mathbb R$ is an open interval. Assume that $f'(x)$ is not $=0$. Show that there is an open interval ...
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1answer
41 views

solve $x+\sin(x)=k$ for $x$ [duplicate]

This question has been proposed to me and thus far it has baffled me: $$ x + \sin(x) = k$$ solve for x. Another way of looking at it is find $f^{-1}(x)$ given that $f(x)=x + \sin(x)$. Wolfram alpha ...
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37 views

Calculate the product ST, and infer from it the inverse of T.

S=\begin{pmatrix} 1/2 & 1/2 & 0\\ 1 & 0 & 0\\ -3/2 & 0 & 1/2 \end{pmatrix} T= \begin{pmatrix} 0 & 1 & 0\\ 2 & -1 & 4\\ 0 & 3 & 2 \end{pmatrix} I ...
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58 views

Use Euclid's Algorithm to find the multiplicative inverse

Use Euclid's Algorithm to find the multiplicative inverse of $13$ in $\mathbf{Z}_{35}$ Can someone talk me through the steps how to do this? I am really lost on this one. Thanks
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1answer
26 views

relationships of symmetric matrices

I came across the following relationships, but I have no idea how to prove them. I would love to know they can be proved. Suppose $X$ and $Y$ are both symmetric matrices, relationship: $$(X + ...
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1answer
19 views

Class of the inverse function

The exercise goes like this: Let $f$ be an invertible function of class $C^k([a,b])$, prove that $f^{-1}$ is of the same class. But wait a second: $f(x) = x^3$ is invertible and of class $C^{\infty}$ ...
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2answers
45 views

Is $\sec^{-1}(\sec(\pi/2)) = \pi/2$?

I think it shouldn't be defined as $\pi/2$ is not in the range of the function $\sec^{-1}(x)$ Wolfram confused me by giving the answer as $\pi/2$ : Link But it mentions on another page that $\pi/2$ ...
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1answer
22 views

Trig and Inverse Trig Function Compositions

Sorry if this is a dumb question, but I honestly tried searching and all I could find was obvious stuff like $\sin(\arcsin(x)) = x$ So what is the logic behind simplifying expressions like this, ...
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1answer
34 views

Is every invertible matrix over an algebraically closed field diagonalisable?

In $\Bbb{R}$ the only invertible matrices (I can think of) that are not diagonalisable are those which stand for a rotation, but in $\Bbb{C}$ this shouldn't be a problem anymore, since rotations can ...
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1answer
58 views

Inverse Function Differential Equation [duplicate]

For the differential equation $$\frac{d}{dx}[y(x)]=y^{(-1)}(x)$$ where $y^{(-1)}(x)$ is the inverse of $y(x)$, find y(x). I gave up on finding the solution analytically pretty quickly and decided ...
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43 views

How to find the inverse of this particular symmetric matrix

Basically, I have a $n \times n$ symmetric matrix, which looks like this: $$ \begin{bmatrix} 1 & \alpha & \cdots & \alpha \\ \alpha & 1 & \cdots &\alpha \\ \vdots &\vdots ...
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How to find the inverse system of a given one

what is the inverse formula of y[n]=x[n]*x[n+1] ? And how can I find the inverse formula/system of a given one in general? I'm having some troubles with this when it comes to some formulas.
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1answer
28 views

How do I show a left inverse of a bounded linear operator on Banach space?

If $A$ is a bounded linear operator on a Banach space X, with a left inverse $A_l^{-1}$, and P is a projection (also on X), how do I show that $A_l^{-1}P$ is also a left inverse of A (i.e. ...
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1answer
70 views

What are some practical uses of functions? [closed]

Functions are basically formal equations that relate a set of inputs to output. What are some practical uses for functions and inverse functions?
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1answer
25 views

Linear algebra proof that AB = On with A invertible only if B = On

$A,B \in Mn(R)$ so that $AB=0n$ and $A$ is an invertible matrix. Proof that $B=0n$ by definition $A$ is invertible so: $\exists C \in Mn : AC=CA=In$ so $A \ne 0n$ Then $AB=0n$ if $B=0n$ Here I can ...
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29 views

Computing the inverse explicitly (real analysis)

I have a function $\ f:\mathbb R\to\mathbb R$ such that $\ f(x,y)=(xe^y,xe^{-y}) $ Let $\ a=(1,0), b=(1,1) $ and let $\ g$ be the continuous inverse of $\ f$ such that $\ g(b)=a$. Compute $\ g$ ...
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33 views

Inverse of Cartan matrix

The Cartan matrix of the root system $A_n$ looks like, denote it by $A'_n$ $$A'_n= \begin{bmatrix} 2 & -1 & 0 & 0&\ldots & 0 \\[0.3em] -1 & 2 & -1 ...
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How to find inverse of $\sin(x) + \sin(2x) = y$?

I was wondering if there were any way to solve the equation $$\sin(x) + \sin(2x) = y$$ in terms of $x$. This of course would allow us to express the inverse for this function on $-\frac{\pi}{4}$ to ...
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Derivatives of component inverse functions

I might have missed the point of the following questions. Anyone kindly give a suggestion? Let $f:\mathbb{R}_\mathbf{x}^3\to\mathbb{R}_\mathbf{y}^3$ and ...
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Singularity in matrix when inverting in Matlab

As data I get a matrix A but in my algorithm I need to work on its inverse. What I do is: C = inv(A) + B; Then in another line I update A. In the next cycles I ...
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1answer
26 views

For any continuous function f(x), how can I split up the function and restrict the domain to find an inverse?

I want to know everything there is to know about inverses for curiosity's sake. I am totally fine finding the inverse of a function where each x maps to a unique y coordinate, but when we get to ...
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1answer
38 views

What does $g(f(x))=x$ imply?

Let $f: X\rightarrow Y$ and $g:Y\rightarrow X$ be functions such that $g(f(x))=x$ for all $x\in X$. (a) Prove that $f$ is injective. (b) Prove that $g$ is surjective. (c) Give an example of a pair ...
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Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = ...
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Self inverting Rings

Would it be possible for a ring to have elements that are their own additive inverses? What I mean is, would it be possible to have a ring $K$ of mathematical objects $A$ such that: $$A+A=i,\;\forall ...
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29 views

Finding inverse using logs

$$ x = \left(\frac{4^y}{-2}\right)^{\frac{1}{3}} $$ i have correct answer of $\:y=\log(4)-2x^3$ i'm lost on steps to obtain the answer. i tried the ...
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Problem inverting a function

I have this function: $$v(t)=\sqrt{\frac F c} \tanh \left(\frac{\sqrt{Fc}}{m} t \right)$$ I can visually see that t=6.3 when v=27.8, so why don't I get t=6.3 upon putting v=27.8 in this supposedly ...
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1answer
81 views

Given its pseudo-inverse, is there a fast way to measure the degree of full-rankness of a nonsquare matrix?

update: I realized the core of question is about ill-conditioning of the matrix (aka Multicollinearity). In a computer, with floating point arithmetic, it is impossible to talk about full-rankness. We ...
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42 views

“Self invertible” group

Let there be an Abelian group with a binary operation $\ast$ on a set $S$. Let such a group respect the following propriety: $$ (X\ast Y)\ast Y = X$$ For any $X$ and $Y$ in $S$. I realize that by ...
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90 views

Inverse of a function

From my text book it says that $f(x)= x^3$ and $f^{-1}(x) = \sqrt[3]{x}$ , which I totally agree with. why does $f(x)= \frac 1 {x-1}$ and $f^{-1}(x)= \frac 1 {x + 1}$ and not equal $f^{-1}(x)= \frac ...
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Finding the inverse of trig functions

I'm supposed to find the inverse of $$f(x) = \cos(x)+x$$ I usually just substitute $x$ for $y$ and then re-arrange. What do I do in this scenario?
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multiplicative inverse in factor ring

If I need to find the multiplicative inverse of an element in some $T[x]/(m)$ factor ring, do I need to solve a diophantine equation to get the solution? Let the element be $f$. Then $fu \equiv 1$ ...
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24 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
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134 views

Conditions for a matrix to be invertible

Let $n \geq m$ and let $C$ be a $n \times m$ full rank matrix, that is $rank(C) =m$. Considering that $D$ is a diagonal positive semidefinite matrix, under which conditions is the $ m \times m$ matrix ...
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1answer
45 views

How to invert a matrix

I would like to disprove the following claim, that seems false to me, finding a counterexample. Let $\mathbf{A}, \mathbf{B} \in \mathbb{R}^{n \times k}$, for $k < n$. Let us assume that $rk(A) ...
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1answer
31 views

Is there a pseudo inverse $X$ such that $ABX=A$?

Question The title pretty much sums it up. I need to find a matrix $X$ such that: $A B X = A$, with $A\in R^{n\times n}$, $\text{rank}(A)=n$, $B\in \mathbb{R}^{n\times m}$ given. The matrix $X$ ...
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1answer
32 views

Number of configurations in a constrained nested loops and configuration back from serial

Consider 4 counters looping the digits 0, 1, 2 to form the various "configurations", like in : ...